Answer
Verified
417k+ views
Hint: Assume two positive integers $n\ and\ \left( n+1 \right)$. Multiply them together. After multiplication, prove that the product obtained can be written in the form $2K$where $'K'$ is any positive integer by using mathematical induction.
Complete step-by-step answer:
According to the question, we have to prove that the product of two positive integers is divisible by 2.
Let us assume two positive integers $n\ and\ \left( n+1 \right)$.
Their product will be $n\times \left( n+1 \right)={{n}^{2}}+n$.
Now, we have to prove that ${{n}^{2}}+n$ is even. Let us use mathematical induction to prove that ${{n}^{2}}+n$ is divisible by 2.
Let,
$\begin{align}
& P\left( n \right)={{n}^{2}}+n \\
& \Rightarrow P\left( 1 \right)=1+1=2 \\
\end{align}$
$P\left( 1 \right)=2$, so $P\left( 1 \right)$ is divisible by 2 and $P\left( 1 \right)$is even.
Let us assume that $P\left( r \right)$ is even.
$i.e.\ ''P\left( r \right)={{r}^{2}}+r''$is even.
So, ${{r}^{2}}+r$ can be written as $2{{K}_{1}}$ where ${{K}_{1}}$ is any positive integer.
${{r}^{2}}+r=2{{K}_{1}}.............\left( 1 \right)$
Now, let us see $P\left( r+1 \right)$,
$\begin{align}
& P\left( r+1 \right)={{\left( r+1 \right)}^{2}}+\left( r+1 \right) \\
& ={{r}^{2}}+2r+1+r+1 \\
\end{align}$
Writing $2r\ as\ r+r$, we will get,
$\begin{align}
& P\left( r+1 \right)={{r}^{2}}+\left( r+r \right)+r+2 \\
& P\left( r+1 \right)=\left( {{r}^{2}}+r \right)+2r+2 \\
& P\left( r+1 \right)=\left( {{r}^{2}}+r \right)+2\left( r+1 \right) \\
\end{align}$
From equation (1), we can write ${{r}^{2}}+r=2{{k}_{1}}$ and we can write $\left( r+1 \right)={{k}_{2}}$.
$\begin{align}
& \Rightarrow P\left( r+1 \right)=2{{K}_{1}}+2{{K}_{2}} \\
& \Rightarrow P\left( r+1 \right)=2\left( {{K}_{1}}+{{K}_{2}} \right) \\
& \Rightarrow P\left( r+1 \right)=2\left( {{K}_{3}} \right)\ \ \ \ \ \ \left[ \text{Assuming }{{K}_{1}}+{{K}_{2}}={{K}_{3}} \right] \\
\end{align}$
As, $P\left( r+1 \right)$ can be written as 2 multiplied by a constant, so $P\left( r+1 \right)$ will be even.
$P\left( 1 \right)$ is even.
And $P\left( r+1 \right)$ is even if $P\left( r \right)$ is even.
So, by mathematical induction, it is proved that ${{n}^{2}}+n$ is even for any positive integer $n$.
Note: In two consecutive numbers, one of the numbers will be odd and another will be even. Product of an odd number and an even number will always be even.
We can write an even number as $2n$ and an odd number as $\left( 2n+1 \right)$.
Then their product,
$\begin{align}
& =\left( 2n \right)\times \left( 2n+1 \right) \\
& =2\times n\times \left( 2n+1 \right) \\
& =2K \\
\end{align}$
So, the product will be even.
Complete step-by-step answer:
According to the question, we have to prove that the product of two positive integers is divisible by 2.
Let us assume two positive integers $n\ and\ \left( n+1 \right)$.
Their product will be $n\times \left( n+1 \right)={{n}^{2}}+n$.
Now, we have to prove that ${{n}^{2}}+n$ is even. Let us use mathematical induction to prove that ${{n}^{2}}+n$ is divisible by 2.
Let,
$\begin{align}
& P\left( n \right)={{n}^{2}}+n \\
& \Rightarrow P\left( 1 \right)=1+1=2 \\
\end{align}$
$P\left( 1 \right)=2$, so $P\left( 1 \right)$ is divisible by 2 and $P\left( 1 \right)$is even.
Let us assume that $P\left( r \right)$ is even.
$i.e.\ ''P\left( r \right)={{r}^{2}}+r''$is even.
So, ${{r}^{2}}+r$ can be written as $2{{K}_{1}}$ where ${{K}_{1}}$ is any positive integer.
${{r}^{2}}+r=2{{K}_{1}}.............\left( 1 \right)$
Now, let us see $P\left( r+1 \right)$,
$\begin{align}
& P\left( r+1 \right)={{\left( r+1 \right)}^{2}}+\left( r+1 \right) \\
& ={{r}^{2}}+2r+1+r+1 \\
\end{align}$
Writing $2r\ as\ r+r$, we will get,
$\begin{align}
& P\left( r+1 \right)={{r}^{2}}+\left( r+r \right)+r+2 \\
& P\left( r+1 \right)=\left( {{r}^{2}}+r \right)+2r+2 \\
& P\left( r+1 \right)=\left( {{r}^{2}}+r \right)+2\left( r+1 \right) \\
\end{align}$
From equation (1), we can write ${{r}^{2}}+r=2{{k}_{1}}$ and we can write $\left( r+1 \right)={{k}_{2}}$.
$\begin{align}
& \Rightarrow P\left( r+1 \right)=2{{K}_{1}}+2{{K}_{2}} \\
& \Rightarrow P\left( r+1 \right)=2\left( {{K}_{1}}+{{K}_{2}} \right) \\
& \Rightarrow P\left( r+1 \right)=2\left( {{K}_{3}} \right)\ \ \ \ \ \ \left[ \text{Assuming }{{K}_{1}}+{{K}_{2}}={{K}_{3}} \right] \\
\end{align}$
As, $P\left( r+1 \right)$ can be written as 2 multiplied by a constant, so $P\left( r+1 \right)$ will be even.
$P\left( 1 \right)$ is even.
And $P\left( r+1 \right)$ is even if $P\left( r \right)$ is even.
So, by mathematical induction, it is proved that ${{n}^{2}}+n$ is even for any positive integer $n$.
Note: In two consecutive numbers, one of the numbers will be odd and another will be even. Product of an odd number and an even number will always be even.
We can write an even number as $2n$ and an odd number as $\left( 2n+1 \right)$.
Then their product,
$\begin{align}
& =\left( 2n \right)\times \left( 2n+1 \right) \\
& =2\times n\times \left( 2n+1 \right) \\
& =2K \\
\end{align}$
So, the product will be even.
Recently Updated Pages
The branch of science which deals with nature and natural class 10 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
Define absolute refractive index of a medium
Find out what do the algal bloom and redtides sign class 10 biology CBSE
Prove that the function fleft x right xn is continuous class 12 maths CBSE
Find the values of other five trigonometric functions class 10 maths CBSE
Trending doubts
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
How fast is 60 miles per hour in kilometres per ho class 10 maths CBSE
What organs are located on the left side of your body class 11 biology CBSE
a Tabulate the differences in the characteristics of class 12 chemistry CBSE
How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Which are the major cities located on the river Ga class 10 social science CBSE
What is BLO What is the full form of BLO class 8 social science CBSE